Uniform and mean approximation by certain weighted polynomials,with applications

LetW(x):= exp(-{tiQ(x})), where, for example, Q(x) is even and convex onR, and Q(x)/logx → ∞ asx → ∞. A result of Mhaskar and Saff asserts that ifa _n =a _n (W) is the positive root of the equation $$n = ({2 mathord{left/ {vphantom {2 pi }} right. kern-nulldelimiterspace} pi })int_0^1 {{{a_n xQ'(a_n x)} mathord{left/ {vphantom {{a_n xQ'(a_n x)} {sqrt {1 - x^2 } }}} right. kern-nulldelimiterspace} {sqrt {1 - x^2 } }}dx,}$$ then, given any polynomialP _n(x) of degree at mostn, the sup norm ofP _n(x)W(a _n x) overR is attained on [-1, 1]. In addition, any sequence of weighted polynomials {p _n (x)W(a _n x)} ₁ ^∞ that is uniformly bounded onR will converge to 0, for ¦x¦>1. In this paper we show that under certain conditions onW, a function g(x) continuous inR can be approximated in the uniform norm by such a sequence {p _n (x)W(a _n x)} ₁ ^∞ if and only if g(x)=0 for ¦x¦? 1. We also prove anL _p analogue for 0W(x)=exp(?|x| ^α ), when α >1. Further applications of our results are upper bounds for Christoffel functions, and asymptotic behavior of the largest zeros of orthogonal polynomials. A final application is an approximation theorem that will be used in a forthcoming proof of Freud's conjecture for |x| ^p exp(?|x| ^α ),α > 0,p > ?1.